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In a right triangle, the sides that form the right angle are called legs. The side opposite the right angle is the hypotenuse. The lengths of the legs and the hypotenuse are related by the Pythagorean theorem. The Pythagorean Theorem 9 3. leg hypotenuse

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NOTE BOOK Words For any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. Algebra a 2 + b 2 = c 2a 2 + b 2 = c 2 Pythagorean Theorem The Pythagorean Theorem 9 3. Numbers 3 2 + 4 2 = 5 2 a = 3 b = 4 c = 5

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The Pythagorean Theorem a 2 + b 2 = c 2 Finding the Length of a Hypotenuse EXAMPLE 1 Find the length of the hypotenuse of a right triangle with leg lengths of 15 inches and 20 inches. 9 3. Pythagorean theorem 225 + 400 = c 2 625 = c 25 = c The length of the hypotenuse is 25 inches. 625 = c 2 Add. Take positive square root of each side. Evaluate square root. ANSWER Substitute 15 for a and 20 for b. Evaluate powers. a 2 + b 2 = c 2 20 2 15 2

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In application problems, you usually need to take only the positive square root. For example, length, speed, and height are positive, so a negative square root would not give a reasonable answer. The Pythagorean Theorem Finding the Length of a Leg EXAMPLE 2 9 3. a 2 + b 2 = c 2 Pythagorean theorem 40,000 + b 2 = 90,000 b = 223.6068 Subtract 40,000 from each side. You are about 224 feet above the water. Parasailing You are parasailing. After getting airborne and reaching cruising speed, you are 200 feet directly behind the boat. How high are you above the water to the nearest foot? SOLUTION b 2 = 50,000 ANSWER Substitute 200 for a and 300 for c. Approximate square root. Take positive square root of each side. Evaluate powers. b = 50,000 a 2 + b 2 = c 2 200 2 300 2

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The Pythagorean Theorem 9 3. Converse of the Pythagorean Theorem The Pythagorean theorem can be written as an if-then statement with two parts. Theorem If a triangle is a right triangle, then a 2 + b 2 = c 2. When you reverse the parts of an if-then statement, the new statement is called the converse of the statement. Converse If a 2 + b 2 = c 2, then the triangle is a right triangle. The converse of a statement may or may not be true. The converse of the Pythagorean theorem is true. You can use the converse of the Pythagorean theorem to decide whether a triangle is a right triangle.

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The Pythagorean Theorem 6 2 + 8 2 = 10 2 ? 36 + 64 = 100 ? Identifying Right Triangles EXAMPLE 3 Use the converse of the Pythagorean theorem to determine whether the triangle with the given side lengths is a right triangle. 9 3. a = 6, b = 8, c = 10 a 2 + b 2 = c 2 100 = 100 A right triangle ANSWER

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The Pythagorean Theorem 6 2 + 8 2 = 10 2 ? 36 + 64 = 100 ? Identifying Right Triangles EXAMPLE 3 Use the converse of the Pythagorean theorem to determine whether the triangle with the given side lengths is a right triangle. 9 3. a = 6, b = 8, c = 10 a 2 + b 2 = c 2 100 = 100 a = 10, b = 12, c = 16 a 2 + b 2 = c 2 244 256 10 2 + 12 2 = 16 2 ? 100 + 144 = 256 ? A right triangle ANSWER Not a right triangle ANSWER

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